Question

Suppose that Fuzzy, a quantum-mechanical duck, lives in a world in which Planck's constant is \(h=1.00 \mathrm{~J}\) s. Fuzzy has a mass of \(0.500 \mathrm{~kg}\) and initially is known to be within a \(0.750-\mathrm{m}\) -wide pond. What is the minimum uncertainty in Fuzzy's speed? Assuming that this uncertainty prevails for \(5.00 \mathrm{~s}\), how far away could Fuzzy be from the pond after 5.00 s?

Short answer

Answer: The minimum uncertainty in Fuzzy's speed is approximately 0.266 m/s, and Fuzzy could potentially be 1.33 m away from the pond after 5 seconds.

Step-by-step solution

Calculate the minimum uncertainty in momentum

First, we need to find the minimum uncertainty in momentum (∆p) using the Heisenberg Uncertainty Principle Equation: ∆x * ∆p ≥ h / (4π) We know that ∆x = 0.750 m, and h = 1.00 J s, so we can rearrange the equation to solve for ∆p: ∆p ≥ h / (4π * ∆x) Now, plug in the given values: ∆p ≥ (1.00 J s) / (4π * 0.750 m) ∆p ≥ 0.133 kg m/s The minimum uncertainty in Fuzzy's momentum is 0.133 kg m/s.

Calculate the minimum uncertainty in speed

We can find the minimum uncertainty in Fuzzy's speed (∆v) using the relation between momentum and speed: ∆p = m * ∆v Here, m (mass) = 0.500 kg, so we can rearrange the equation to solve for ∆v: ∆v = ∆p / m Now, plug in the given values: ∆v = (0.133 kg m/s) / (0.500 kg) ∆v ≈ 0.266 m/s The minimum uncertainty in Fuzzy's speed is approximately 0.266 m/s.

Calculate how far away Fuzzy could be from the pond after 5.00 s

Assuming the speed uncertainty prevails for 5.00 s, we can calculate the maximum distance Fuzzy could travel away from the pond during this time: Distance (d) = ∆v * t Here, t (time) = 5.00 s, so plug in the given values: d = (0.266 m/s) * (5.00 s) d ≈ 1.33 m Thus, Fuzzy could potentially be 1.33 m away from the pond after 5.00 s.

Key concepts

Quantum Mechanics

Quantum mechanics represents a fundamental pillar of modern physics, explaining the behavior of particles on an atomic and subatomic scale. It diverges significantly from classical physics, which cannot accurately predict phenomena at these incredibly small scales. In the world of quantum mechanics, particles such as electrons and photons exhibit wave-particle duality, meaning they can behave both as particles and as waves.

This field of physics also introduces the concept of quantization, which indicates that certain properties, such as energy, can only occur in specific, discrete amounts. These quantum states form the basis for understanding the complex and often non-intuitive behaviors of particles. Concepts like superposition and entanglement challenge our understanding of location and state, suggesting that particles can exist in multiple states or locations until they are measured. As a result, much of the quantum world is governed by probabilities and uncertainties, rather than deterministic outcomes.

Furthermore, this field has paved the way for advancements in technology, including the development of semiconductors, lasers, and quantum computers. It's a reminder that the behavior of particles at the quantum level can have a profound impact on our everyday lives, from the way we compute to the functioning of solar cells.

Momentum Uncertainty

The concept of momentum uncertainty is closely tied to the Heisenberg Uncertainty Principle, which is a cornerstone of quantum mechanics. According to the principle, it is impossible to simultaneously measure both the position (represented by \(\Delta x\)) and the momentum (represented by \(\Delta p\)) of a particle with absolute precision. The more precisely one of these values is known, the less precisely the other can be known. This inherent uncertainty is not due to experimental shortcomings but is a fundamental property of quantum systems.

The relationship is quantified by the inequality \(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\), where \(h\) is Planck's constant. In our example with Fuzzy the quantum-mechanical duck, this principle determines the minimum uncertainty in the duck's speed based on the level of our knowledge about its position. If Fuzzy's position within the pond is known to be within a 0.750 m range, there is a corresponding uncertainty in its momentum, and thus its speed. This conceptual linkage between measurement and existence is fundamental to quantum mechanics, and it prompts deeper questions about the nature of reality itself.

Planck's Constant

Planck's constant (\(h\)) is a physical constant that plays a critical role in quantum mechanics. It was first introduced by physicist Max Planck and is fundamental in the formulation of quantum theory. The constant has dimensions of action which is energy multiplied by time, typically measured in joule-seconds (J s).

Its value is incredibly small, approximately \(6.62607015 \times 10^{-34} \text{J s}\), indicating the scale at which quantum effects become significant. The constant is named after Max Planck, who used it as a proportionality constant to describe the energy (\(E\)) of a photon in relation to its frequency (\(u\)) through the equation \(E = hu\).

Moreover, Planck's constant is the quantum of electromagnetic action that relates a photon's energy to its frequency. It also sets the scale for the Heisenberg Uncertainty Principle, which defines limits to how precisely the position and momentum of a quantum object can be known simultaneously. Thus, Planck's constant is not only a marker of quantum scales but is also integral in calculating phenomena that involve quantization, as seen in exercises involving quantum mechanics like the one with Fuzzy.